410 XV NORMED ALGEBRAS AND SPECTRAL THEORY
17. Let E be an infinite-dimensional separable Hilbert space and N a normal continuous
operator on E.
(a) Show that E is the Hilbert sum EI + E2 of two infinite-dimensional subspaces,
each of which is stable under TV and TV*. (Reduce to the case of a simple normal
operator M^(lc); with the notation of (15.11.9), observe that if M is a closed subset of
Sp(AO, then E(M) can be finite-dimensional only if M is a finite set for which each
point has yit-measure ^=0. Then distinguish two cases, according as there exist infinitely
many points of measure ^0 or not; in the second case, use Problem 3(b) of Section
13.18.)
(b) Deduce from (a) that there exists a decomposition of E as the Hilbert sum of an
infinite sequence (En) of infinite-dimensional subspaces, each of which is stable under
N and N*.
18. (a) Let E be a separable Hilbert space which is the Hilbert sum of an infinite family
(£„)„ 6 z of infinite-dimensional subspaces. There exists a unitary operator S on E
such that S(En) = En + i for all n e Z. Show that if P (resp. Q) is the operator which
is equal to S1 ~ 2n (resp. S~ 2n) on En for each n, then P2 = Q2 = 1 E , and S = PQ.
(b) Deduce from (a) and from Problem 17 that every unitary operator on an
infinite-dimensional separable Hilbert space is the product of four involutory unitary
operators.
(c) Let o> be a complex cube root of unity and let U be the homothety with ratio
CD on E, which is a unitary operator. Show that I/ is not the product of three involutory
unitary operators. (In general, in a group G, if t is in the center of G, and if there exist
x, y, z in G such that / ~ xyz and x2 = y2 — z2 = 1, then also / = yzx, t2 = xyxy and
19. (a) Let E be an infinite-dimensional separable Hilbert space and let (en)n>o be a
Hilbert basis of E. The continuous operator K such that V'en = en + 1 for all
n^>Qis called the one-sided shift operator; it is an isometry of E onto the hyperplane
orthogonal to e0; its spectrum is the disk |£| ^ 1 and contains no eigenvalue of V
(Section 11.1, Problem 4); its approximative point-spectrum (Problem 9) is the circle
U : |f | = 1. The spectrum of the adjoint operator K* is also the disk|£| <£ 1, and every
£ such that |£| < 1 is an eigenvalue of K*.
(b) Let T be a continuous operator on E which is an isometry of E onto a (necessarily
closed) subspace T(E). Show that there exists a decomposition of E as the Hilbert sum
of subspace L and an at most denumerable family (Ft)iei of subspaces, where L and
each Ft is stable under T and are such that (1) 7*1 L is unitary and (2) each F( is infinite-
dimensional and T\Fi is the one-sided shift operator, for a suitably chosen ortho-
normal basis. (Consider the orthogonal supplement N of T(E), and show that E is
the Hilbert sum of the TW(N) (n ^ 0) and L = fl Tn(E).)
n
(c) Deduce from (a) and (b) that if T is any nonunitary isometry of E onto a sub-
space of E, then Sp(T) is the unit disk |£| <; 1, and that \\T~ U\\ = 2 for all unitary
operators U. (Observe that ||r— U\\ = \\U*T— 1E|| and that U*T is not unitary, so
that the point £ = - 1 belongs to its spectrum.)
20. (a) Let E be a separable Hilbert space, T a continuous operator, and C a compact
operator on E. Show that the points of Sp(7t+ C) which do not belong to Sp(T)
are eigenvalues of T+ C. (Reduce to the case where £ = 0 is such a point and observe
that, if T is bijective, we may write T-f- C= T(1E + T~*C), where T~*C is compact;
if _i e Sp(7T-1C), it follows that -1 is an eigenvalue of T~1C.)uch that (jitw) converges vaguely to a measure which does not belong to M£(X, v).